AP

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

f(x)=\dfrac{x^2-4}{x-2}

. (i) Show that direct substitution of

x=2

f(x)

leads to the indeterminate form

0/0

. (ii) Explain briefly why the existence of the indeterminate form

0/0

x=2

does not automatically imply that

\displaystyle \lim_{x\to 2}f(x)

fails to exist. (iii) Name the algebraic property of polynomials which allows common factors of the numerator and denominator to be cancelled for

x\neq 2

, thereby enabling the limit to be evaluated.